Quantum interference between dark-excitons and zone-edged acoustic phonons in few-layer WS2

Fano resonance which describes a quantum interference between continuum and discrete states, provides a unique method for studying strongly interacting physics. Here, we report a Fano resonance between dark excitons and zone-edged acoustic phonons in few-layer WS2 by using the resonant Raman technique. The discrete phonons with large momentum at the M-point of the Brillouin zone and the continuum dark exciton states related to the optically forbidden transition at K and Q valleys are coupled by the exciton-phonon interactions. We observe rich Fano resonance behaviors across layers and modes defined by an asymmetry-parameter q: including constructive interference with two mirrored asymmetry Fano peaks (weak coupling, q > 1 and q < − 1), and destructive interference with Fano dip (strong coupling, ∣q∣ < < 1). Our results provide new insight into the exciton-phonon quantum interference in two-dimensional semiconductors, where such interferences play a key role in their transport, optical, and thermodynamic properties.

By fitting and analyzing the high-frequency Raman spectra of few-layer WS2, we found that there are no obvious peak broadening or asymmetry lineshape for 2LA(M), ′ ( 2g 1 ) and ′ 1g ( ′ 1 ) modes, as shown in Fig. S4 to S7. It suggests that there are no Fano resonance for ′ ( 2g 1 ) and ′ 1g ( ′ 1 ) phonon modes (the optical phonon modes at Γ point). The vibration of these modes is the relative motion inside the primitive cell atoms. Usually, the treatment of the interactions between electrons and these phonon modes should employ the polaron-related theory, different from this work's physical picture. Probably this is why we do not observe the quantum interference phenomenon for high frequency optical branch phonon modes.
We note that the energy of 594 nm excitation (2.09 eV) wavelength is slightly higher than the X0 in few-layer WS2. In that case, the high-frequency A1g mode is greatly enhanced, implying that the out-of-plane A1g mode strongly couples to the bright A excitons. This can be understood that the propagation (polarization) direction of bright A exciton is out-of-plane (in-plane), which can effectively couple with out-of-plane A1g phonons.

Section III, Symmetry Analysis of phonon modes and excitons Phonon modes at  point and M point:
For the shear (S) modes at Γ point, the phonon symmetry is 2g 2 . The corresponding Raman tensor is doubly degenerate.   Under the resonant excitation with dark A exciton, the Raman selection rules are determined by the exciton-phonon interaction, rather than by the Raman tensor of phonon modes [1], as a result, these Raman inactive modes also can be observed.

Exciton symmetry:
The exciton symmetry is composed of symmetry of the envelop function (internal orbital symmetry) and the electron (conduction band) and hole (valence band) symmetry. The representation can be given by Γ = Γ ⊗ Γ ⊗ Γ * [2]. Here we only need to consider the symmetry of 1s state in our case.
For dark A exciton, the symmetry is Γ 1 = Γ 9 ⊗ Γ 7 * = Γ 9 ⊗ Γ 8 = Γ 4 in 3ℎ symmetry at K point. The polarization of dark A exciton is out-of-plane (Z polarization, corresponding to a z-dipole transition), which is in-plane dipole transition forbidden for normal incidence configuration [2][3][4]. We should note that the dark A exciton is slightly mixed with bright A exciton, which could be brightened by a large in-plane magnetic field [5][6][7].
Meanwhile, the dark B exciton state is the upper band one (Fig. S1). The forming of a continuum state by exciton population is more difficult than that for the dark A exciton. As a result, under such resonance excitation conditions, the Fano resonance of shear modes basically vanishes, and the Fano resonance of zone-edged acoustic modes at the M point is greatly weakened.
For the dark exciton with finite momentum, the electron at Q valley holds σ symmetry while the hole at K valley holds a 3ℎ symmetry. The dark excitons are momentum forbidden.

The resonance situation with dark B excitons
We note that a weak Fano intensity of ZA(M) and LA(M) can be resolved with 514 nm excitation. This result can be understood by considering that this excitation energy is close to B exciton. Hence, the scattering picture that we mentioned above (see Fig.   S1) results in the observed ZA(M) and LA(M). However, the nature of dark B exciton states (upper band and zero oscillator strength [3] and the above optical bandgap excitation significantly weaken such process.